Does a Drunken Master Monk have to attack with one of their flurry of blows attacks to activate the disengage benefit?
No. The benefits of an ability are not its costs/requirements.
Drunken Technique states:
At 3rd level, whenever you use Flurry of Blows, you gain the benefit of the Disengage action, and your walking speed increases by 10 feet until the end of the current turn.
So all you have to do is use Flurry of Blows. The rule for Flurry of Blows states:
Flurry of Blows
Immediately after you take the Attack action on your turn, you can spend 1 ki point to make two unarmed strikes as a bonus action.
Using Flurry of Blows requires you to spend 1 Ki. It is this payment of Ki that activates Flurry of Blows and grants the benefits of Disengage (and 2 attacks).
Put another way, if you use Step of Wind (which allows you to Dash or Disengage as a bonus action and increases your jump distance), you are not required to jump (or even move, for that matter).
If a DM is not inclined to just let you simply choose not to attack with the 2 attacks granted by Flurry of Blows (which seems like a silly insistence to me), consider this: You are able to attack a location, not just creatures or objects:
- Choose a target. Pick a target within your attack's range: a creature, an object, or a location.
That location could be any space, including one where no creature exists, which is functionally identical to not making an attack unless there's some reason you absolutely don't want to make an attack at all (eg, something bad happens if you attack).
But then, if you don't want to attack but still disengage as a bonus action, you could just use that 1 ki towards Step of the Wind instead:
Step of the Wind
You can spend 1 ki point to take the Disengage or Dash action as a bonus action on your turn, and your jump distance is doubled for the turn.
From the description for Drunken Master
At 3rd level, you learn how to twist and turn quickly as part of your Flurry of Blows. Whenever you use Flurry of Blows, you gain the benefit of the Disengage action, and your walking speed increases by 10 feet until the end of the current turn.
So you do have to use Flurry of Blows in order to activate the Disengage.